Classification of bilinear maps with radical of codimension 2

Abstract

Let $\mathbb{V}$ be an $n$-dimensional linear space over an algebraically closed base field. We provide a classification, up to equivalence, of all of the bilinear maps $f:\mathbb{V}\times \mathbb{V}\to \mathbb{V}$ such that $dim(rad(f))=n-2$. This is equivalent to give a complete classification (up to isomorphism) of all $n$-dimensional algebras with annihilator of dimension $n-2$ or, in other words, a classification of the annihilator extensions of all $2$-dimensional algebras.